The only service to Dublin, from Liverpool, is with P&O but they don't carry foot passengers on that route.
Stena Line operate from Liverpool (Birkenhead) to Belfast. The day sailings leave each port at 1030, with the night sailings leaving at 2230. It's a long trip, taking around 8 hours. Day sailings cost £17.50. Night sailings cost £32.50. To get a cabin you'd need to add £20 on to those fares.
So you could get a long day in Belfast by taking the overnight sailings in both directions but it would cost you £65 (or £105 with a cabin in each direction).
A short trip to the Isle of Man is far easier but, because there's no early morning sailing out of Liverpool, you'd need to stay overnight. The 1115 sailing arrives at 1345, giving you the afternoon and evening on the island. (There's also a 1900 sailing, arriving at 2130). You could then have the following morning on the island, before catching the 1500 sailing (arriving back at 1730).
Timetable here:
http://www.steam-packet.com/en/Timetables
(NB: That page defaults to the current timetable. For sailings after 26th of this month, remember to set the correct dates).
Fares here:
http://www.steam-pack...abulousFootlooseFares
(The day trip fares are irrelevant as, with the current timetable, they only apply to travel from Heysham).
With many ferry companies merging (e.g. Norfolk Line, DFDS and Stena) and others going out of business completely, the above appears to be a complete list of services from Liverpool.
If you'd consider travelling by air, you can fly from John Lennon Airport to Belfast City (FlyBe), Belfast International (easyJet), Isle of Man (FlyBe and easyJet) and Dublin (Ryanair). Ryanair, for example, are currently offering a £35.98 return fare (including taxes, charges and their infamous 'administration fees') for travel to Dublin on many dates next month.
Chris